The multiplier of the interval [formula omitted] for the Dunkl transform on the real line

Abstract

We study the boundedness of the multiplier of the interval [ − 1 , 1 ] for the Dunkl transform of order α ⩾ − 1 / 2 on weighted L p spaces, with 1 < p < ∞ . In particular, we get that it is bounded from L p ( R , | x | 2 α + 1 d x ) into itself if and only if 4 ( α + 1 ) / ( 2 α + 3 ) < p < 4 ( α + 1 ) / ( 2 α + 1 ) .